58 Connect Words and Algebra Teaching Skill 58 Objective Connect words and algebra. Point out to students that an algebraic expression (or equation) is simply a mathematical way of writing a phrase or sentence. Ask: What is the difference between an expression and an equation? (An expression does not have an equal sign while an equation does.) Tell students that associating key words with the correct operation is critical to being able to make the connection between words and algebra. Review with students the table of key words. Provide some real life examples. Start with the following two: (1)If your age is 10 years more than your sister s, and your sister is 12, how do you find your age? (Add 12 plus 10.) If you receive $10 per week allowance for 8 weeks, how do you find the total amount you received? (Multiply 10 times 8.) Have students provide additional examples. Review the lawn cutting example with students and then have them complete the exercises. PRACTICE ON YOUR OWN In exercises 1 6, students write algebraic expressions and equations. CHECK Determine that students know how to write algebraic expressions and equations. Students who successfully complete the Practice on Your Own and Check are ready to move on to the next skill. COMMON ERRORS Students may not know the key words associated with operations, and as a result may use the wrong operation. Students who made more than 1 error in the Practice on Your Own, or who were not successful in the Check section, may benefit from the Alternative Teaching Strategy. Alternative Teaching Strategy Objective Connect words and algebra. Materials needed: multiple copies of the flashcards shown below (index cards work nicely), blank index cards Have students work in pairs. Distribute copies of the flashcards and have students shuffle the cards and divide them in half so that each student has six cards. 3x 1 n 5 x 7 1 n 2 C 3(25) t 6(2) 2 C 12 2.50 P 2(8) 2w A 1 2 (6)(h) I (500)(2)r A 50 10w S 600 50w Review with students key words and the operations with which the words are associated. Tell students they are going to make up phrases or situations that match the algebraic expressions and equations on the cards. Give the following example: 5n 3. Ask: How would you verbalize 5n? (5 times a number) And 3? (increased by 3) Write the phrase on the board: 5 times a number increased by 3. Point out that variations of this phrase are also correct, such as 3 more than 5 times a number. Next, write the equation C 40 5(d ) on the board. Remind students that variables typically represent something that makes sense. For example, C might represent cost and d number of days. Or, provide the following scenario: There are 40 cookies in a bag and Tom eats 5 cookies per day. C 40 5(d ) could be the number of cookies remaining at the end of any given day. Instruct students to take turns making up situations to match the cards they have. As an extension of the exercise, have students make flashcards and continue. 127 Holt Algebra 2
Name Date Class 58 Connect Words and Algebra To connect words and algebra, you must understand the operations involved and how to represent them. Key words are helpful in determining the operations. Key Words a number; an unknown quantity Twice, three times, etc. sum; more than; increased by difference; less than; decreased by each; per is; equals Operation or Representation any variable, such as x or n multiplication (2n, 3n, etc.) addition ( ) subtraction ( ) multiplication Example: Jared must cut 6 lawns over the weekend. Each of the lawns takes 2 hours to cut. Write an equation representing the total time t to cut all 6 lawns. Answer: Since each lawn takes 2 hours, multiply 2 times the number of lawns to get the total time: t 6(2). Practice on Your Own 1. Write an expression that represents the quantity 5 more than a number. 2. Write a phrase that could be modeled by the expression x 15. 3. John bought 3 CDs and 2 DVDs. Each CD costs $9.95, and each DVD costs $14.98. Write an equation representing the total cost C. 4. A triangle has sides of length 7, 10, and s. Write an equation representing the perimeter P of the triangle. 5. The value of a painting begins at $12,000 and increases by $500 per year. Write an equation representing the value V of the painting at the end of any given year y. 6. David has 56 baseball cards of which he sells 3 cards per week. Write an equation representing the number of cards n he has left at the end of any given week w. Check 7. Write an expression that represents a number decreased by 6. 8. Tina bought 6 plates and 2 glasses. Each plate costs $6.99, and each glass costs $22.98. Write an equation representing the total cost C. 9. Joseph opens a checking account with $400. Each month he adds $150 to the account. Write an equation representing the total amount A in the account at the end of any given month m. 128 Holt Algebra 2
68 Solve One-Step Equations Teaching Skill 68 Objective Solve one-step equations. Explain to students that inverse operations are operations that undo each other. Direction students attention to the first example. Ask a volunteer to read the addition equation x 5 15. Ask: What operation is being done to the variable? (5 is being added to it.) How can you undo this? (Subtract 5.) So, what is the inverse operation of addition? (subtraction) Work through the next example. Stress that since subtraction is the inverse operation of addition, addition is the inverse operation of subtraction. Go through a similar process to explain the multiplication and division examples. Remind students that they should be careful when working with negatives. Ask: If a multiplication equation contains 2x, what do you divide by? ( 2) Does that mean the answer is negative? (not necessarily) Give examples if time permits. Have students complete the practice exercises. PRACTICE ON YOUR OWN In exercises 1 12, students solve one-step equations that require addition, subtraction, multiplication, or division. CHECK Determine that students know how to solve one-step equations. Students who successfully complete the Practice on Your Own and Check are ready to move on to the next skill. COMMON ERRORS Students may forget to use an inverse operation when solving equations. Students who made more than 2 errors in the Practice on Your Own, or who were not successful in the Check section, may benefit from the Alternative Teaching Strategy. Alternative Teaching Strategy Objective Solve one-step equations. Materials needed: multiple copies of the flashcards shown below (index cards work nicely) n 3 9 n 12 4 n 6 2 n 8 12 n 1 5 n 15 11 5n 50 7n 42 9n 45 n 6 4 n 15 2 n 100 5 Tell students they are going to practice identifying inverse operations and then solving one-step equations. Remind students that inverse operations undo each other. Have students work in pairs. Give each pair of students one set of equation cards. Instruct students to mix the cards up and leave them face down. Students should take turns drawing one card at a time. The student who draws the card should face it toward their partner. The partner should read the equation aloud, identify the operation being done to the variable, and then identify how to undo the operation. The partner does not need to solve the equation. The student holding the card should then turn the card around and read it, confirm their partner s answers, and then solve the equation. Students should repeat the exercise until all the cards have been drawn and all the equations solved. An extension of this exercise is to have students make their own index cards which should include addition, subtraction, multiplication and division equations. 147 Holt Algebra 2
Name Date Class 68 Solve One-Step Equations To solve a one-step equation, do the inverse of whatever operation is being done to the variable. Remember, because it is an equation, what is done to one side of the equation must also be done to the other side. Solve an addition equation using subtraction. x 5 15 5 5 x 10 Solve a multiplication equation using division. 7x 42 7x 42 7 7 x 6 Solve a subtraction equation using addition. x 8 3 8 8 x 5 Solve a division equation using multiplication. x 3 12 12 x 3 12 12 x 36 Practice on Your Own Solve. 1. m 5 9 2. h 3 3. 6x 54 4. b 15 25 6 5. 4y 12 6. k 9 3 7. p 7 2 8. t 3 7 9. x 1 10. 5 h 16 11. 12x 24 12. r 2 9 4 Check Solve. 13. 3x 15 14. c 11 1 15. d 9 5 16. s 6 5 17. z 2 17 18. w 4 12 19. 10b 120 20. x 99 100 148 Holt Algebra 2
49 Percent Problems Teaching Skill 49 Objective Find the percent of a number. Review with students how to change a percent to a decimal. Point out that if the percent is a one digit number (e.g. 7%), they will need to add a zero as a placeholder (e.g. 7% 0.07). Ask: What do you have to remember when multiplying by a decimal? (The number of decimal places in the final product must equal the number of decimal places in the factors.) Tell students that percent problems can easily be converted into equations. Review the translation chart with students. Ask: How can you represent an unknown quantity? (You can use any variable, such as n or x.) Work through the example with students. Point out that the percent must be converted to a decimal before multiplying. Have students complete the exercises. Encourage the students to estimate the products before multiplying so they will know if their answers make sense. PRACTICE ON YOUR OWN In exercises 1 4, students find the product of a decimal and a whole number. In exercises 5 10, students find the percent of a number. CHECK Determine that students know how to find the percent of a number. Students who successfully complete the Practice on Your Own and Check are ready to move on to the next skill. COMMON ERRORS Students may forget to change the percent to a decimal before multiplying. Students who made more than 2 errors in the Practice on Your Own, or who were not successful in the Check section, may benefit from the Alternative Teaching Strategy. Alternative Teaching Strategy Objective Find the percent of a number. Explain to students that they can find percents using proportions. Write the following words on the board: cents, century, centipede, and centimeter. Ask: What does cent mean? (100) Ask: What does percent mean? (per 100) Ask: Is 25% the same as 25 100? (Yes) Remind students that a proportion is two ratios that are equal to each other. Write on the board this proportion: is % of 100. Tell students that they can use this proportion to solve percent problems. For example, have students consider the following question: What is 25% of 200? Have students underline what is once, 25% twice, and 200 three times. What is 25% of 200? Explain that what (or n since what is an unknown quantity) should be substituted for is ; 200 should be substituted for of ; and 25 should be substituted for %. n 25 200 100 Ask: How do you solve a proportion? (Find the cross products.) What are the cross products? (100n and 5000). Write the equation 100n 5000 on the board. Ask: How do you solve for the variable? (Divide both sides of the equation by 100.) Solve the equation: 100n 5000 ; n = 50. So, 50 is 25% of 200. 100 100 Have students use this strategy to answer the following questions: What is 80% of 50? (40); What is 15% of 400? (60); What is 35% of 120? (42); and What is 90% of 440? (396) 109 Holt Algebra 2
Name Date Class 49 Percent Problems Multiplying by percents: Step 1: Change the percent to a decimal by dropping the % symbol and moving the decimal point two places to the left. Step 2: Multiply using rules for decimal multiplication. Translating a percent problem into an equation: Rewrite the percent as a decimal and then use the translations at the right to rewrite the problem as an equation. Example: What is 30% of 90? n 0.30 90 Word Mathematical Translation what an unknown quantity, such as n or x is equals or of multiplication or or or ( ) n 27 Practice on Your Own Multiply. 1. 0.25 72 2. 0.15 60 3. 0.20 1400 4. 0.06 500 Answer each question. 5. What is 12% of 50? 6. What is 70% of 30? 7. What is 22% of 150? 8. What is 10% of 450? 9. What is 50% of 168? 10. What is 65% of 4000? Check Multiply. 11. 0.08 250 12. 0.35 60 13. 0.40 600 14. 0.75 480 Answer each question. 15. What is 3% of 200? 16. What is 20% of 115? 17. What is 45% of 180? 18. What is 95% of 300? 110 Holt Algebra 2
21 Convert Units of Measure Teaching Skill 21 Objective Convert units of measure. Point out to students that converting between units of measure always involves either multiplication or division. Ask: Is it sometimes difficult to remember when to multiply and when to divide? (Most students will probably say yes.) Introduce the concept and review the definition of a conversion factor. Emphasize that you do not have to remember whether to multiply or divide if you use a conversion fact; it depends on the numerator and denominator of the fraction. Have students look at the common conversion factors. Ask: Why does a conversion factor equal 1? (Because the measure in the numerator is equal to the measure in the denominator even though the units are different) Instruct students to consider the example. Explain that the correct conversion factor is 1 yd (not 3 ft ) because you need feet in the 3 ft 1 yd denominator to cancel with the feet in the problem. Work through the example. PRACTICE ON YOUR OWN In exercises 1 9, students convert from one unit of measure to another. CHECK Determine that students know how to convert units of measure. Students who successfully complete the Practice on Your Own and Check are ready to move on to the next skill. COMMON ERRORS Students may multiply when they should divide, or divide when they should multiply. Students who made more than 2 errors in the Practice on Your Own, or who were not successful in the Check section, may benefit from the Alternative Teaching Strategy. Alternative Teaching Strategy Objective Convert units of measure. Materials needed: multiple copies of the game cards shown below; copy of page 2 of this lesson (Common Conversion Factors) 4 ft 46 in. 1.3 yd 1200 sec 35 min 0.5 hr 2 gal 9 qt 15 pt Have students review the chart of common conversion factors. Review with students how conversion factors are used when converting from one unit of measure to another. Give each student a set of game cards. Have students briefly look at the cards and take note of the symbols on the cards. Explain that the footprint represents length; the clock represents time; and the pitcher represents capacity. Have students shuffle their cards (face down). Tell students they are going to play Speed. Explain to students that when you say Go, they are to flip their cards over, separate them into categories (length, time, and capacity), and put the cards in each category in order from least to greatest. Students should use scratch paper to convert to whichever units they believe will help them the most. The student who correctly puts all the cards in order wins. Answers: Length: 46 in., 1.3 yd, 4 ft; Time: 1200 sec, 0.5 hr, 35 min; and Capacity: 15 pt, 2 gal, 9 qt An extension of this exercise is to have students create their own cards, using different units than the ones provided. 53 Holt Algebra 2
Name Date Class 21 To convert from one unit of measure to another, you can use a conversion factor and multiply. Definition: a conversion factor is a fraction equal to 1 whose numerator and denominator have different units. 1 min 60 sec 1 day 24 hr Convert Units of Measure Common Conversion Factors Length Capacity Weight 1 ft 12 in. 1 yd 3 ft 1 mi 5280 ft or 60 sec 1 min or 24 hr 1 day or 12 in. 1 ft or 3 ft 1 yd or 5280 ft 1 mi 1 yr 12 mo Time 1 hr 60 min 1 wk 7 days or 12 mo 1 yr Example: Convert 12 feet to yards. 12 ft 1 yd 12 yd 4 yd 3 ft 3 Practice on Your Own Convert each unit of measure. or 60 min 1 hr of 7 days 1 wk 1 pt 2 c or 2 c 1 pt 1 qt 2 pt 1 gal 4 qt or 2 pt 1 qt or 4 qt 1 gal 1 m 100 cm or 100 cm 1 m 1 km 1000 m Metric 1 lb 16 oz 1 T 2000 lb 1 m 1000 mm or 1000 m 1 km or 16 oz 1 lb or 2000 lb 1 T or 1000 mm 1 m Note: All metric units have the same conversion factors based on the prefix. 1. 30 months to years 2. 48 inches to feet 3. 3 pounds to ounces 4. 650 centimeters to meters 5. 1.5 days to hours 6. 16 quarts to gallons 7. 7000 pounds to tons 8. 2.2 liters to milliliters 9. 7 yards to feet Check Convert each unit of measure. 10. 9.2 meters to centimeters 11. 2.5 hours to minutes 12. 22 pints to quarts 13. 4500 milligrams to grams 14. 56 ounces to pounds 15. 4.5 years to months 54 Holt Algebra 2
54 Absolute Value Teaching Skill 54 Objective Find the absolute value of numbers and expressions. Review with students the definition of absolute value at the top of the page. Have students read the statement that begins with 6. Ask: What other number is 6 units from 0? ( 6) Does that mean that 6 is also 6? (Yes) Review the general rule with students. Ask: Can the statement n 2 ever be true? (No) Why not? (The general rule tells us that the absolute value of any nonzero number is always a positive value.) Point out that this makes sense, since distance is always positive as well. Review with students the steps for evaluating an absolute value expression. Point out that the expression inside the absolute value symbols should be evaluated first, then find the distance from zero. Review each of the examples with students and have them complete the practice exercises. PRACTICE ON YOUR OWN In exercises 1 12, students find the absolute value of numbers and expressions. CHECK Determine that students know how to find absolute value. Students who successfully complete the Practice on Your Own and Check are ready to move on to the next skill. COMMON ERRORS Students may change negative numbers to positive numbers inside the absolute value symbols before performing the indicated operation. Students who made more than 2 errors in the Practice on Your Own, or who were not successful in the Check section, may benefit from the Alternative Teaching Strategy. Alternative Teaching Strategy Objective Find the absolute value of numbers and expressions. Draw the table below on the board and instruct the students to duplicate it on their paper. Negative Value Distance from 0 1 (1) 1 2 (2) 2 3 (3) 3 4 (4) 4 5 (5) 5 Positive Value Remind students that the absolute value of a number is the distance between that number and zero. Tell students they are going to complete the center column by using a number line. Instruct students to draw a number line and label it from 5 to 5. For each of the values in the table, instruct students to count the number of jumps (the distance) between that value and zero. Encourage students to make conjectures about the relationship between n and n. (They are equal.) Draw the table below on the board and instruct students to duplicate it. Expression Simplified Distance from 0 7 4 ( 3 ) (3) 5 12 ( 7 ) (7) 10 6 ( 4 ) (4) 1 (1) 12 (12) 15 (15) For the first three rows, instruct students to complete the second column for each of the expressions. Then they should use number lines to find the distance to zero and complete the third column. For the last three rows, have students make up expressions that would result in column 2, and then complete column 3. (Answers may vary in column 1.) 119 Holt Algebra 2
Name Date Class 54 Absolute Value Definition: The absolute value of a number is the distance between that number and zero on a number line. 6 is read as the absolute value of 6 and means the distance between 6 and 0 on a number line. Distance of 6 3 0 5 6 6 General rule: The absolute value of any nonzero number is always a positive value. To evaluate an expression that contains an absolute value: Step 1: Evaluate the expression inside the absolute value symbols (as if they were parentheses). Step 2: Take the absolute value of the final result (make it positive). Example 1: 7 7 Example 2: 14 9 5 5 Example 3: 0.5 0.5 Example 4: 20 23 3 3 Practice on Your Own Find the absolute value of each expression. 1. 15 2. 8 3. 0.4 4. 1.19 5. 25 15 6. 18 22 7. 0.25 1 8. 4.6 3.9 9. 8 14 10. 9 2 11. 5 12 7 12. 23 7 42 Check Find the absolute value of each expression. 13. 11 14. 2.3 15. 50 40 16. 100 75 17. 80 93 18. 2.5 2.5 19. 5.2 4.1 20. 11 14 2 120 Holt Algebra 2